And Edwards equation

Doi, M. and Edwards, S.F., 1978. Dynamics of concentrated polymer systems 1. Brownian motion in equilibrium state, 2. Molecular motion under flow, 3. Constitutive equation and 4. Rheological properties. J. Cheni. Soc., Faraday Trans. 2 74, 1789, 1802, 1818-18.32. 

Our approach in this chapter is to alternate between experimental results and theoretical models to acquire familiarity with both the phenomena and the theories proposed to explain them. We shall consider a model for viscous flow due to Eyring which is based on the migration of vacancies or holes in the liquid. A theory developed by Debye will give a first view of the molecular weight dependence of viscosity an equation derived by Bueche will extend that view. Finally, a model for the snakelike wiggling of a polymer chain through an array of other molecules, due to deGennes, Doi, and Edwards, will be taken up. 

Both the Edwards equation and Pearson s HSAB concept take as primary determinants of nucleophilicity the polarizability and basicity. A two-term equation of Bartoli and Todesco" uses these ideas also, but as a measure of polarizability the... 

Equation of State Calculations by Fast Computing Machines Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller and Edward Teller Journal of Chemical Physics 21 (1953) 1087... 

The position of aniline in the above reactivity order deserves special comment. Aniline is less basic than pyridine by a relatively small factor, 0.65 pA units, but is appreciably more polarizable it then seems likely that the inverted order of reactivity is caused by the polarizability term in accordance with Edwards equation. If this is correct, in the reactivity order piperidine > aniline > pyridine, inversion with respect to basicity appears to result from an abnormally high reactivity of aniline rather than from a particularly low reactivity of pyridine. This view differs from that based on relative steric requirements of the reagents, but other factors besides basicity and polarizability may well contribute to the quantitative experimental picture. 

Bircumshaw and Edwards [1029] showed that the rate of nickel formate decomposition was sensitive to reactant disposition, being relatively greater for the spread reactant, a—Time curves were sigmoid and obeyed the Prout—Tompkins equation [eqn. (9)] with values of E for spread and aggregated powder samples of 95 and 110 kJ mole-1, respectively. These values are somewhat smaller than those subsequently found [375]. The decreased rate observed for packed reactant was ascribed to an inhibiting effect of water vapour which was most pronounced during the early stages. 

In Eq. (10-17), parameters a and b measure the sensitivity of the reaction to these nucleophilic parameters. Since H measures proton basicity and En the electron-donation ability, this treatment models nucleophilicity as a combination of electron loss and electron pair donation. The Edwards equation is thus an oxibase scale of nucleophilic reactivity. Table 10-5 summarizes the nucleophilic parameters. 

Figure 7. age vs. growth band age for three coral sub-samples, all younger than 200 years old (after Edwards 1988 and Edwards et al. 1988). All three points lie on a 1 1 line indicating that the °Th ages are accurate and that initial is negligible, justifying the use of Equation (1) to determine °Th age. 

At the high polymer concentration used in plasticized systems the viscosity of amorphous polymer is given by the modified Rouse theory at low molecular weight, M - 2Mr [from equation (47)] and by the modified Doi-Edwards equation at high molecular weight. In the first case... 

Recently, there have been a number of significant developments in the modeling of electrolyte systems. Bromley (1), Meissner and Tester (2), Meissner and Kusik (2), Pitzer and co-workers (4, ,j5), and" Cruz and Renon (7j, presented models for calculating the mean ionic activity coefficients of many types of aqueous electrolytes. In addition, Edwards, et al. (8) proposed a thermodynamic framework to calculate equilibrium vapor-liquid compositions for aqueous solutions of one or more volatile weak electrolytes which involved activity coefficients of ionic species. Most recently, Beutier and Renon (9) and Edwards, et al.(10) used simplified forms of the Pitzer equation to represent ionic activity coefficients. 

Recently, the Pitzer equation has been applied to model weak electrolyte systems by Beutier and Renon ( ) and Edwards, et al. (10). Beutier and Renon used a simplified Pitzer equation for the ion-ion interaction contribution, applied Debye-McAulay s electrostatic theory (Harned and Owen, (14)) for the ion-molecule interaction contribution, and adoptee) Margules type terms for molecule-molecule interactions between the same molecular solutes. Edwards, et al. applied the Pitzer equation directly, without defining any new terms, for all interactions (ion-ion, ion-molecule, and molecule-molecule) while neglecting all ternary parameters. Bromley s (1) ideas on additivity of interaction parameters of individual ions and correlation between individual ion and partial molar entropy of ions at infinite dilution were adopted in both studies. In addition, they both neglected contributions from interactions among ions of the same sign. 

VAN AKEN et al. 0) and EDWARDS et al. (2) made clear that two sets of fundamental parameters are useful in describing vapor-liquid equilibria of volatile weak electrolytes, (1) the dissociation constant(s) K of acids, bases and water, and (2) the Henry s constants H of undissociated volatile molecules. A thermodynamic model can be built incorporating the definitions of these parameters and appropriate equations for mass balance and electric neutrality. It is complete if deviations to ideality are taken into account. The basic framework developped by EDWARDS, NEWMAN and PRAUSNITZ (2) (table 1) was used by authors who worked on volatile electrolyte systems the difference among their models are in the choice of parameters and in the representation of deviations to ideality. 

The kinetic equation for the distribution function f(r, a t) must include all these effects. Doi and Edwards [4,99] proposed for it the generalized Smoluchowski equation... 

The model described by equations (3.42)-(3.45) is valid for equilibrium situations. For chain in a flow, one ought to define displacements of the particles under flow and to consider the average values (3.44) to depend on the velocity gradient (Doi and Edwards 1986). McLeish and Milner (1999) considered mechanism of reptation motion of branched macromolecules of different architecture. 

These are exactly the known results (Doi and Edwards 1986, p. 196). The time behaviour of the equilibrium correlation function is described by a formula which is identical to formula for a chain in viscous liquid (equation (4.34)), while the Rouse relaxation times are replaced by the reptation relaxation times. In fact, the chain in the Doi-Edwards theory is considered as a flexible rod, so that the distribution of relaxation times naturally can differ from that given by equation (4.36) the relaxation times can be close to the only disentanglement relaxation time r[ep. 

One can see that the approximation of the theory, based on the linear dynamics of a macromolecule, is not adequate for strongly entangled systems. One has to introduce local anisotropy in the model of the modified Cerf-Rouse modes or use the model of reptating macromolecule (Doi and Edwards 1986) to get the necessary corrections (as we do in Chapters 4 and 5, considering relaxation and diffusion of macromolecules in entangled systems). The more consequent theory can be formulated on the base of non-linear dynamic equations (3.31), (3.34) and (3.35). 

One has no results for this case derived consequently from the basic equations (7.6) with local anisotropy. Instead, to find conformational relaxation equation, we shall use the Doi-Edwards model, which approximate the large-scale conformational changes of the macromolecule due to reptation. The mechanism of relaxation in the Doi-Edwards model was studied thoroughly (Doi and Edwards 1986 Ottinger and Beris 1999), which allows us to write down the simplest equation for the conformational relaxation for the strongly entangled systems... 

The concept of soft and hard acids and bases (7), which is in effect an extension of the Chatt-Ahrland classification (2) of A and B metals, is applied in the 1963 paper by Pearson (7) particularly to equilibria involving mainly inorganic systems. This paper follows an earlier discussion by Edwards and Pearson (3) of the Swain-Edwards equation (4) (1) for nucleophilic reactivity. 

An ingenious treatment of cellulose was discovered by Charles Cross and Edward Bevan in England in 1892. It involved first preparing a chemical derivative called cellulose xanthate in a process that is conceptually no different from converting cellulose into other derivatives such as cellulose acetate or cellulose nitrate. What made this different, however, is that xan-thates are reactive chemical intermediates that can be converted easily into still different compounds, or returned to the starting material, in this case cellulose. See Equation 3. 

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